Finite Group Databasebeta

← #7792

Subgroups — #7792

Order 256.

187 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2726C211yesmaximal, characteristic
2#2099C211yesmaximal, characteristic
3#2083C211yesmaximal, characteristic
4#2083C211yesmaximal, characteristic
5#1271C211yesmaximal, characteristic
6#1271C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C2×#125—21no—
9C2×#107—21no—
10C2×#107—21no—
11#398—21no—
12#398—21no—
13#398—21no—
14#389—21no—
15#389C2211yescharacteristic
16#389C2211yescharacteristic
17#389C2211yescharacteristic
18#387—21no—
19#384C2211yescharacteristic
20#340—21no—
21#340—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×C8—21no—
28C22×C8—21no—
29C22×C8—21no—
30#125—42no—
31#125—42no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C42—21no—
49C2×C42C2311yescharacteristic
50#107—42no—
51#107—42no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—41no—
65C22×C4C2×D411yescharacteristic
66C22×C4C2×D411yescharacteristic
67C22×C4C2×D411yescharacteristic
68C2×C8—22no—
69C2×C8—22no—
70C2×C8—41no—
71C2×C8—41no—
72C2×C8—41no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8—41no—
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—81no—
99C4⋊C4D812yes—
100C4⋊C4Q1612yes—
101C42—41no—
102C42—42no—
103C23#12011yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—81no—
129C2×C4—81no—
130C2×C4—81no—
131C2×C4C2×D811yescharacteristic
132C2×C4C2×Q1611yescharacteristic
133C2×C4#13511yescharacteristic
134C2×C4AGL(1,ℤ/8)11yescharacteristic
135C2×C4AGL(1,ℤ/8)11yescharacteristic
136C2×C4#13711yescharacteristic
137C8—42no—
138C8—42no—
139C8—81no—
140C22#44711yescharacteristic
141C22#44911yescharacteristic
142C22#45111yescharacteristic
143C22#45411yescharacteristic
144C22#45611yescharacteristic
145C22#45611yescharacteristic
146C22#45811yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—42no—
154C4—42no—
155C4—81no—
156C4—81no—
157C4—81no—
158C4—81no—
159C4—161no—
160C2#160911yescharacteristic
161C2#161011yescharacteristic
162C2#161711yescharacteristic
163C2#162511yescharacteristic
164C2#164811yescharacteristic
165C2#164911yescharacteristic
166C2#166311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#27262
#20992
#20832
#12712